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Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences of $f_1$ and $f_2$.

For instance, suppose that we have maps $g_X : C^*(X_1) \to C^*(X_2)$ and $g_Y: C^*(Y_1) \to C^*(Y_2)$ of singular cochain complexes (not necessarily coming from topological maps), such that the following diagram commutes: $\require{AMScd}$ \begin{CD} C^*(Y_1) @>f_1^*>> C^*(X_1)\\ @Vg_YVV @VVg_XV\\ C^*(Y_2) @>f_2^*>> C^*(X_2). \end{CD}

Does this give rise to map between the Leray spectral sequences of $f_1$ and $f_2$ ?

More generally, I'm looking for a reference where the construction of the Leray spectral sequence is done entirely. So far I have seen two constructions: one rather categorical with Grothendieck spectral sequence, and the other one using the Cech-DeRham double complex (Bott-Tu). Both constructions are problematic for me because I'm dealing with quite general topological spaces (so Cech-DeRham is not relevant), and I work with maps at the level of cochain complexes, so Grothendieck spectral sequence doesn't really help.

Thanks a lot.

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  • $\begingroup$ I think you will get a map of Leray sequences when your vertical maps are ring homomorphisms. $\endgroup$ – Phil Tosteson Jul 8 '18 at 15:18
  • $\begingroup$ Thanks for your comment @Phil. Could you be more specific ? Why would ring homomorphisms give maps between spectral sequences ? What ring ? $\endgroup$ – BrianT Jul 8 '18 at 17:34
  • $\begingroup$ Cochains form a ring under the cup product. For the Leray-Serre spectral sequence, when the base spaces are simply connected, you should be able to recover the spectral sequences purely algebraically from the ring structure on cochains. I don't know a reference, but maybe you can try Mcleary's users guide to spectral sequences. $\endgroup$ – Phil Tosteson Jul 8 '18 at 23:41
  • $\begingroup$ The reason why ring structure helps is this: To get a map between Leray-Serre sequences, you at least need your maps of chain complexes to induce maps on the cohomology of the homotopy fibers. When the base space is simply connected, you can use the ring structure to figure out what the cohomology of the homotopy fiber is a la Eilenberg Moore. $\endgroup$ – Phil Tosteson Jul 8 '18 at 23:51
  • $\begingroup$ Thanks, the spectral sequences that I consider here are Leray spectral sequences, associated with continuous maps which are not necessarily fibrations. Nevertheless, $Y_1$ and $Y_2$ are indeed simply connected in my example. Does this still work ? $\endgroup$ – BrianT Jul 9 '18 at 15:11

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